Statistics 2
swells36
SBReview.docxUse a calculator to verify that Σx = 24.3, Σx2 = 87.47, Σy = 54.2, Σy2 = 468.56 and Σxy = 194.66. Compute r. (Round to 3 decimal places.)
Use a calculator to verify that Σx = 1.951, Σx2 = 0.550, Σy = 30.4, Σy2 = 150.72 and Σxy = 8.806. Compute r. (Round your answer to three decimal places.)
You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some calves to add to the Bar-S herd. How much should a healthy calf weigh? Let x be the age of the calf (in weeks), and let y be the weight of the calf (in kilograms).
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x |
3 |
2 |
11 |
16 |
26 |
36 |
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y |
44 |
54 |
71 |
100 |
150 |
200 |
Complete parts (a) through (e), given Σx = 94, Σy = 619, Σx2 = 2362, Σy2 = 82,393, Σxy = 13,721, and r ≈ 0.991.
(b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)
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Σx = |
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Σy = |
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Σx2 = |
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Σy2 = |
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Σxy = |
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r = |
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(c) Find x, and y. Then find the equation of the least-squares line = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)
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x |
= |
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y |
= |
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= |
+ x |
Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.)
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r2 = |
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explained |
% |
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unexplained |
% |
(f) The calves you want to buy are 22 weeks old. What does the least-squares line predict for a healthy weight? (Round your answer to two decimal places.)
Let x be the age of a licensed driver in years. Let y be the percentage of all fatal accidents (for a given age) due to failure to yield the right of way. For example, the first data pair states that 5% of all fatal accidents of 37-year-olds are due to failure to yield the right of way.
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x |
37 |
47 |
57 |
67 |
77 |
87 |
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y |
5 |
8 |
10 |
17 |
32 |
43 |
Complete parts (a) through (e), given Σx = 372, Σy = 115, Σx2 = 24814, Σy2 = 3351, Σxy = 8475, and r ≈ 0.949.
(b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)
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Σx = |
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Σy = |
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Σx2 = |
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Σy2 = |
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Σxy = |
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r = |
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(c) Find x, and y. Then find the equation of the least-squares line = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)
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x |
= |
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y |
= |
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= |
+ x |
Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.)
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r2 = |
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explained |
% |
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unexplained |
% |
(f) Predict the percentage of all fatal accidents due to failing to yield the right of way for 75-year-olds. (Round your answer to two decimal places.) %
After a large fund drive to help the Boston City Library, the following information was obtained from a random sample of contributors to the library fund. Using a 1% level of significance, test the claim that the amount contributed to the library fund is independent of ethnic group.
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Number of People Making Contribution |
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Ethnic Group |
$1-50 |
$51-100 |
$101-150 |
$151-200 |
Over $200 |
Row Total |
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A |
80 |
65 |
54 |
31 |
21 |
251 |
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B |
92 |
47 |
74 |
28 |
23 |
264 |
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C |
85 |
64 |
53 |
36 |
28 |
266 |
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D |
106 |
82 |
67 |
56 |
29 |
340 |
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Column Total |
363 |
258 |
248 |
151 |
101 |
1121 |
(a) What is the level of significance?
(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)
What are the degrees of freedom?
The types of raw materials used to construct stone tools found at an archaeological site are shown below. A random sample of 1486 stone tools were obtained from a current excavation site.
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Raw Material |
Regional Percent of Stone Tools |
Observed Number of Tools as Current excavation Site |
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Basalt |
61.3% |
932 |
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Obsidian |
10.6% |
167 |
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Welded Tuff |
11.4% |
155 |
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Pedernal chert |
13.1% |
184 |
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Other |
3.6% |
48 |
Use a 1% level of significance to test the claim that the regional distribution of raw materials fits the distribution at the current excavation site.
(a) What is the level of significance?
(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)
What are the degrees of freedom?
Let x represent the number of mountain climbers killed each year. The long-term variance of x is approximately σ2 = 136.2. Suppose that for the past 7 years, the variance has been s2 = 116.2. Use a 1% level of significance to test the claim that the recent variance for number of mountain-climber deaths is less than 136.2. Find a 90% confidence interval for the population variance.
(a) What is the level of significance?
(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.) What are the degrees of freedom?
(c) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.)
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lower limit |
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upper limit |
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(d)
An economist wonders if corporate productivity in some countries is more volatile than in other countries. One measure of a company's productivity is annual percentage yield based on total company assets. A random sample of leading companies in France gave the following percentage yields based on assets.
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4.5 |
5.4 |
3.8 |
3.5 |
2.7 |
3.5 |
2.8 |
4.4 |
5.7 |
3.4 |
4.1 |
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6.8 |
2.9 |
3.2 |
7.2 |
6.5 |
5.0 |
3.3 |
2.8 |
2.5 |
4.5 |
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Use a calculator to verify that the sample variance is s2 ≈ 1.997 for this sample of French companies. Another random sample of leading companies in Germany gave the following percentage yields based on assets.
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3.2 |
3.3 |
3.5 |
4.6 |
5.6 |
5.5 |
5.0 |
5.4 |
3.2 |
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3.5 |
3.7 |
2.6 |
2.8 |
3.0 |
3.0 |
2.2 |
4.7 |
3.2 |
Use a calculator to verify that s2 ≈ 1.139 for this sample of German companies. Test the claim that there is a difference (either way) in the population variance of percentage yields for leading companies in France and Germany. Use a 5% level of significance. How could your test conclusion relate to the economist's question regarding volatility (data spread) of corporate productivity of large companies in France compared with companies in Germany?
(a) What is the level of significance?
(b) Find the value of the sample F statistic. (Use 2 decimal places.) What are the degrees of freedom?
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dfN |
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dfD |
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Professor Fair believes that extra time does not improve grades on exams. He randomly divided a group of 300 students into two groups and gave them all the same test. One group had exactly 1 hour in which to finish the test, and the other group could stay as long as desired. The results are shown in the following table. Test at the 0.01 level of significance that time to complete a test and test results are independent.
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Time |
A |
B |
C |
F |
Row Total |
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1 h |
22 |
44 |
61 |
14 |
141 |
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Unlimited |
19 |
45 |
84 |
11 |
159 |
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Column Total |
41 |
89 |
145 |
25 |
300 |
(i) Give the value of the level of significance.
(ii) Find the sample test statistic. (Round your answer to two decimal places.)
A machine that puts corn flakes into boxes is adjusted to put an average of 15.4 ounces into each box, with standard deviation of 0.21 ounce. If a random sample of 14 boxes gave a sample standard deviation of 0.32 ounce, do these data support the claim that the variance has increased and the machine needs to be brought back into adjustment? (Use a 0.01 level of significance.)
(i) Give the value of the level of significance.
(ii) Find the sample test statistic. (Round your answer to two decimal places.)
A sociologist is studying the age of the population in Blue Valley. Ten years ago, the population was such that 21% were under 20 years old, 10% were in the 20- to 35-year-old bracket, 34% were between 36 and 50, 24% were between 51 and 65, and 11% were over 65. A study done this year used a random sample of 210 residents. This sample is given below. At the 0.01 level of significance, has the age distribution of the population of Blue Valley changed?
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Under 20 |
20 - 35 |
36 - 50 |
51 - 65 |
Over 65 |
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29 |
25 |
68 |
66 |
22 |
(i) Give the value of the level of significance.
(ii) Find the sample test statistic. (Round your answer to two decimal places.)
Two processes for manufacturing large roller bearings are under study. In both cases, the diameters (in centimeters) are being examined. A random sample of 21 roller bearings from the old manufacturing process showed the sample variance of diameters to be
s2 = 0.238.
Another random sample of 29 roller bearings from the new manufacturing process showed the sample variance of their diameters to be
s2 = 0.13.
Use a 5% level of significance to test the claim that there is a difference (either way) in the population variances between the old and new manufacturing processes.
(i) Give the value of the level of significance.
(ii) Find the sample test statistic. (Round your answer to two decimal places.)
